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1973 : A two-dimensional chaotic quasi-conservative map

Christophe LETELLIER
31/12/2010
Igor Gumowski & Christian Mira
JPG - 8.3 ko
Christian Mira

In the early 70s, Igor Gumowski and Christian Mira studied families of maps obtained from a dissipative perturbation of conservative maps issued from the studies on the problem of ``stochastic’’ instability in accelerators and storage rings [1]. Among these two families, there is a two-dimensional quasi-conservative map which may be written as


\left\{
  \begin{array}{l}
   \displaystyle
    x_{n+1} = y_n + \mu x_n + \frac{2 (1-\mu) x_n^2}{1 + x_n^2} + \alpha (1-\beta y_n^2}) y_n \\[0.4cm]
    \displaystyle
    y_{n+1} = -x_n + \mu x_{n+1} + \frac{2 (1-\mu) x_{n+1}^2}{1 + x_{n+1}^2} 
  \end{array}
\right.

where parameters values were  \alpha = 0.005 and  \mu = 1/8,  \beta being the bifurcation parameter. This two-dimensional map was published in 1973  [2] and republished in 1980 [3]. Two chaotic attractors were presented as being ``stochastic’’ in 1973 (Fig. 1).

(a)  \beta = -1

(b)  \beta = -10

Fig. 1. Chaotic solution to the two-dimensional quasi-conservative map. Initial conditions x0=0.01 and y0=0. These two pictures were similar to those published in 1973 and 1980.

[1] C. Mira, Memories of the early days of chaos theory in The Chaos avant-garde, World Scientific Series on Nonlinear Science A, 39, pp. 95-197, 2000.

[2] J. Bernussen, Liu Hsu & C. Mira, Quelques exemples du second ordre, Collected preprints of Colloque International du CNRS, 229, Transformations ponctuelles et applications (Toulouse, September 1973), Proceedings Editions du CNRS (Paris), pp. 195-226, 1976.

[3] I. Gumowski & C. Mira, Recurrences and discrete dynamic systems : An introduction, Lecture Notes in Mathematics, 809, 1980.

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